\documentclass{article}
\usepackage{amsmath,amssymb,graphicx}

\newcommand{\R}[0]{\mathbb{R}}
\newcommand{\C}[0]{\mathbb{C}}

\newcommand{\mypar}[1]{{\bf #1.}}

% Title.
% ------
\title{A fast algorithm for sampled approximation of chance-constrained problem}

\begin{document}
%
\maketitle
%
\mypar{Motivation} 
Generic chance-constrianed optimization problem (CCP) is largely intractable except 
for some special cases for two reasons: the feasible region is generally not convex
and evaluating solution feasibility requires multi-dimensional integration. 
Recent approaches ~\cite{Luedtke2008, Luedtke2008a, Luedtke2010} involves Sample Average Approximation (SAA)
and reformulation of the approximation problem (SAA problem) as a mixed-integer program (MIP). This framework is promising to provide good solution to large instances of generic CCP ~\cite{Luedtke2008a}. It remains computational challenge to solve the SAA problem efficiently. 

\mypar{Related work} 
Big-m coefficients are used in the MIP reformulation of the SAA problem, which caused weak LP
relaxation to slow down the Branch-and-Bound algorithm used by the MIP solvers. Strenghened valid inequalities are discussed in ~\cite{Luedtke2008, Luedtke2008a, Luedtke2010} to avoid the big-m coefficient. Decomposition method is developed by ~\cite{Luedtke2010} to deal with a two-stage CCP. (These existing method can be used for comparision with our method, but it requires studying and extra coding which is of lower priorityfor the work of this project.) 

\mypar{Method}
We propose a variant of traditional Branch-and-Bound algorithm (A(aptive)BB) to directly tackle the difficulty arised from relatively large values of big-m coeffcient in the MIP formulation. Aimed at providing stronger LP relaxation, the main idea of the algorithm is to start with relatively small big-m values, though producing suboptimal solution, and then increase them iteratively until true optimal solution is garanteed to be found. CPLEX C API along with its advanced features such as callback functions will be used to implement the algorithm. Instances and data sets will be generated following the discussion in ~\cite{Luedtke2008, Luedtke2008a}. 
  
\mypar{Evaluation}
Although theoretical proof will be given to validate the ABB algorithm, obersavation of emperical behavior will be emphasized through various numerical experiments. Test problems with random variable of discrete distribution and finite support will be firstly investigated, since the true solution can be easily found through alternative ways. More complicated problems will be considered later in the work.
 
\bibliography{proposal}
\bibliographystyle{plain}
\end{document}

